Computational BioMedical Informatics
description
Transcript of Computational BioMedical Informatics
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Computational BioMedical Informatics
SCE 5095: Special Topics Course
Instructor: Jinbo BiComputer Science and Engineering Dept.
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Course Information
Instructor: Dr. Jinbo Bi – Office: ITEB 233– Phone: 860-486-1458– Email: [email protected]
– Web: http://www.engr.uconn.edu/~jinbo/– Time: Mon / Wed. 2:00pm – 3:15pm – Location: CAST 204– Office hours: Mon. 3:30-4:30pm
HuskyCT– http://learn.uconn.edu– Login with your NetID and password– Illustration
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Review of previous classes
Finish Linear Discriminant Analysis
Support Vector Machines
Paper reviews on many technical topics
Start to discuss unsupervised learning
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What is Cluster Analysis?
Finding groups of objects such that the objects in a group will be similar (or related) to one another and different from (or unrelated to) the objects in other groups
Inter-cluster distances are maximized
Intra-cluster distances are
minimized
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What is not Cluster Analysis?
Supervised classification– Have class label information
Simple segmentation– Dividing students into different registration groups
alphabetically, by last name
Results of a query– Groupings are a result of an external specification
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Notion of a Cluster can be Ambiguous
How many clusters?
Four Clusters Two Clusters
Six Clusters
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Types of Clusterings
A clustering is a set of clusters
Important distinction between hierarchical and partitional sets of clusters
Partitional Clustering– A division data objects into non-overlapping subsets
(clusters) such that each data object is in exactly one subset
Hierarchical clustering– A set of nested clusters organized as a hierarchical
tree
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Partitional Clustering
Original Points A Partitional Clustering
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Hierarchical Clustering
p4p1
p3
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p4 p1
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p2 p4p1 p2 p3
p4p1 p2 p3
Traditional Hierarchical Clustering
Non-traditional Hierarchical Clustering Non-traditional Dendrogram
Traditional Dendrogram
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Other Distinctions Between Sets of Clusters
Exclusive versus non-exclusive– In non-exclusive clusterings, points may belong to
multiple clusters.– Can represent multiple classes or ‘border’ points
Fuzzy versus non-fuzzy, probability vs non-probability– In fuzzy clustering, a point belongs to every cluster
with some weight between 0 and 1– Weights must sum to 1– Probabilistic clustering has similar characteristics
Partial versus complete– In some cases, we only want to cluster some of the
data Heterogeneous versus homogeneous
– Cluster of widely different sizes, shapes, and densities
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Types of Clusters
Well-separated clusters
Center-based clusters
Contiguous clusters
Density-based clusters
Property or Conceptual
Described by an Objective Function
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Types of Clusters: Well-Separated
Well-Separated Clusters: – A cluster is a set of points such that any point in a
cluster is closer (or more similar) to every other point in the cluster than to any point not in the cluster.
3 well-separated clusters
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Types of Clusters: Center-Based
Center-based– A cluster is a set of objects such that an object in a
cluster is closer (more similar) to the “center” of a cluster, than to the center of any other cluster
– The center of a cluster is often a centroid, the average of all the points in the cluster, or a medoid, the most “representative” point of a cluster
4 center-based clusters
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Types of Clusters: Contiguity-Based
Contiguous Cluster (Nearest neighbor or Transitive)– A cluster is a set of points such that a point in a
cluster is closer (or more similar) to one or more other points in the cluster than to any point not in the cluster.
8 contiguous clusters
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Types of Clusters: Density-Based
Density-based– A cluster is a dense region of points, which is
separated by low-density regions, from other regions of high density.
– Used when the clusters are irregular or intertwined, and when noise and outliers are present.
6 density-based clusters
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Types of Clusters: Conceptual Clusters
Shared Property or Conceptual Clusters– Finds clusters that share some common property or
represent a particular concept. .
2 Overlapping Circles
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Types of Clusters: Objective Function Clusters Defined by an Objective Function
– Finds clusters that minimize or maximize an objective function.
– Enumerate all possible ways of dividing the points into clusters and evaluate the `goodness' of each potential set of clusters by using the given objective function. (NP Hard)
– Can have global or local objectives. Hierarchical clustering algorithms typically have local objectives Partitional algorithms typically have global objectives
– A variation of the global objective function approach is to fit the data to a parameterized model.
Parameters for the model are determined from the data. Mixture models assume that the data is a ‘mixture' of a number of
statistical distributions.
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Types of Clusters: Objective Function …
Map the clustering problem to a different domain and solve a related problem in that domain– Proximity matrix defines a weighted graph, where the
nodes are the points being clustered, and the weighted edges represent the proximities between points
– Clustering is equivalent to breaking the graph into connected components, one for each cluster.
– Want to minimize the edge weight between clusters and maximize the edge weight within clusters
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Characteristics of the Input Data Are Important
Type of proximity or density measure– This is a derived measure, but central to clustering
Sparseness– Dictates type of similarity– Adds to efficiency
Attribute type– Dictates type of similarity
Type of Data– Dictates type of similarity– Other characteristics, e.g., autocorrelation
Dimensionality Noise and Outliers Type of Distribution
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Clustering Algorithms
K-means and its variants
Hierarchical clustering
Density-based clustering
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K-means Clustering
Partitional clustering approach Each cluster is associated with a centroid (center point) Each point is assigned to the cluster with the closest
centroid Number of clusters, K, must be specified The basic algorithm is very simple
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K-means Clustering – Details Initial centroids are often chosen randomly.
– Clusters produced vary from one run to another. The centroid is (typically) the mean of the points in the
cluster. ‘Closeness’ is measured by Euclidean distance, cosine
similarity, correlation, etc. K-means will converge for common similarity measures
mentioned above. Most of the convergence happens in the first few
iterations.– Often the stopping condition is changed to ‘Until relatively
few points change clusters’ Complexity is O( n * K * I * d )
– n = number of points, K = number of clusters, I = number of iterations, d = number of attributes
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Two different K-means Clusterings
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Optimal Clustering
Original Points
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Importance of Choosing Initial Centroids
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Importance of Choosing Initial Centroids
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Evaluating K-means Clusters
Most common measure is Sum of Squared Errors (SSE)– For each point, the error is the distance to the nearest cluster– To get SSE, we square these errors and sum them.
– x is a data point in cluster Ci and mi is the representative point for cluster Ci
can show that mi corresponds to the center (mean) of the cluster
– Given two clusters, we can choose the one with the smallest error
– One easy way to reduce SSE is to increase K, the number of clusters
A good clustering with smaller K can have a lower SSE than a poor clustering with higher K
K
i Cxi
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xmdistSSE1
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Importance of Choosing Initial Centroids …
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Importance of Choosing Initial Centroids …
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Problems with Selecting Initial Points
If there are K ‘real’ clusters then the chance of selecting one centroid from each cluster is small. – Chance is relatively small when K is large– If clusters are the same size, n, then
– For example, if K = 10, then probability = 10!/1010 = 0.00036
– Sometimes the initial centroids will readjust themselves in ‘right’ way, and sometimes they don’t
– Consider an example of five pairs of clusters
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10 Clusters Example
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Starting with two initial centroids in one cluster of each pair of clusters
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10 Clusters Example
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10 Clusters Example
Starting with some pairs of clusters having three initial centroids, while other have only one.
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10 Clusters Example
Starting with some pairs of clusters having three initial centroids, while other have only one.
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Solutions to Initial Centroids Problem
Multiple runs– Helps, but probability is not on your side
Sample and use hierarchical clustering to determine initial centroids
Select more than k initial centroids and then select among these initial centroids– Select most widely separated
Postprocessing Bisecting K-means
– Not as susceptible to initialization issues
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Handling Empty Clusters
Basic K-means algorithm can yield empty clusters
Several strategies– Choose the point that contributes most to SSE– Choose a point from the cluster with the
highest SSE– If there are several empty clusters, the above
can be repeated several times.
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Updating Centers Incrementally
In the basic K-means algorithm, centroids are updated after all points are assigned to a centroid
An alternative is to update the centroids after each assignment (incremental approach)– Each assignment updates zero or two
centroids– More expensive– Introduces an order dependency– Never get an empty cluster– Can use “weights” to change the impact
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Pre-processing and Post-processing
Pre-processing– Normalize the data– Eliminate outliers
Post-processing– Eliminate small clusters that may represent outliers– Split ‘loose’ clusters, i.e., clusters with relatively high
SSE– Merge clusters that are ‘close’ and that have relatively
low SSE– Can use these steps during the clustering process
ISODATA
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Bisecting K-means
Bisecting K-means algorithm– Variant of K-means that can produce a partitional
or a hierarchical clustering
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Bisecting K-means Example
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Limitations of K-means
K-means has problems when clusters are of differing – Sizes– Densities– Non-globular shapes
K-means has problems when the data contains outliers.
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Limitations of K-means: Differing Sizes
Original Points K-means (3 Clusters)
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Limitations of K-means: Differing Density
Original Points K-means (3 Clusters)
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Limitations of K-means: Non-globular Shapes
Original Points K-means (2 Clusters)
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Overcoming K-means Limitations
Original Points K-means Clusters
One solution is to use many clusters.Find parts of clusters, but need to put together.
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Overcoming K-means Limitations
Original Points K-means Clusters
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Overcoming K-means Limitations
Original Points K-means Clusters
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Summary
Introduced the types of clusterings, and the types of clusters
Discussed k-means (k-medoids) Various issues of k-means and some heuristic
strategies to overcome them Fundamentally, k-means assumes Gaussian
distribution of the same round shape among all clusters
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Hierarchical Clustering
Produces a set of nested clusters organized as a hierarchical tree
Can be visualized as a dendrogram– A tree like diagram that records the
sequences of merges or splits
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Strengths of Hierarchical Clustering
Do not have to assume any particular number of clusters– Any desired number of clusters can be
obtained by ‘cutting’ the dendogram at the proper level
They may correspond to meaningful taxonomies– Example in biological sciences (e.g., animal
kingdom, phylogeny reconstruction, …)
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Hierarchical Clustering
Two main types of hierarchical clustering– Agglomerative:
Start with the points as individual clusters At each step, merge the closest pair of clusters until only one
cluster (or k clusters) left
– Divisive: Start with one, all-inclusive cluster At each step, split a cluster until each cluster contains a point
(or there are k clusters)
Traditional hierarchical algorithms use a similarity or distance matrix– Merge or split one cluster at a time
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Agglomerative Clustering Algorithm
More popular hierarchical clustering technique Basic algorithm is straightforward
1. Let each data point be a cluster2. Compute the proximity matrix3. Repeat4. Merge the two closest clusters5. Update the proximity matrix6. Until only a single cluster remains
Key operation is the computation of the proximity of two clusters
– Different approaches to defining the distance between clusters distinguish the different algorithms
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Starting Situation
Start with clusters of individual points and a proximity matrix
p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
. Proximity Matrix
...p1 p2 p3 p4 p9 p10 p11 p12
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Intermediate Situation
After some merging steps, we have some clusters
C1
C4
C2 C5
C3
C2C1
C1
C3
C5
C4
C2
C3 C4 C5
Proximity Matrix
...p1 p2 p3 p4 p9 p10 p11 p12
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Intermediate Situation
We want to merge the two closest clusters (C2 and C5) and update the proximity matrix.
C1
C4
C2 C5
C3
C2C1
C1
C3
C5
C4
C2
C3 C4 C5
Proximity Matrix
...p1 p2 p3 p4 p9 p10 p11 p12
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After Merging
The question is “How do we update the proximity matrix?”
C1
C4
C2 U C5
C3? ? ? ?
?
?
?
C2 U C5C1
C1
C3
C4
C2 U C5
C3 C4
Proximity Matrix
...p1 p2 p3 p4 p9 p10 p11 p12
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How to Define Inter-Cluster Similarity
p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
.
Similarity?
MIN MAX Group Average Distance Between Centroids Other methods driven by an objective
function– Ward’s Method uses squared error
Proximity Matrix
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How to Define Inter-Cluster Similarity
p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
.Proximity Matrix
MIN MAX Group Average Distance Between Centroids Other methods driven by an objective
function– Ward’s Method uses squared error
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How to Define Inter-Cluster Similarity
p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
.Proximity Matrix
MIN MAX Group Average Distance Between Centroids Other methods driven by an objective
function– Ward’s Method uses squared error
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How to Define Inter-Cluster Similarity
p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
.Proximity Matrix
MIN MAX Group Average Distance Between Centroids Other methods driven by an objective
function– Ward’s Method uses squared error
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How to Define Inter-Cluster Similarity
p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
.Proximity Matrix
MIN MAX Group Average Distance Between Centroids Other methods driven by an objective
function– Ward’s Method uses squared error
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Cluster Similarity: MIN or Single Link
Similarity of two clusters is based on the two most similar (closest) points in the different clusters– Determined by one pair of points, i.e., by one
link in the proximity graph.
I1 I2 I3 I4 I5I1 1.00 0.90 0.10 0.65 0.20I2 0.90 1.00 0.70 0.60 0.50I3 0.10 0.70 1.00 0.40 0.30I4 0.65 0.60 0.40 1.00 0.80I5 0.20 0.50 0.30 0.80 1.00 1 2 3 4 5
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Hierarchical Clustering: MIN
Nested Clusters Dendrogram
1
2
3
4
5
6
12
3
4
5
3 6 2 5 4 10
0.05
0.1
0.15
0.2
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Strength of MIN
Original Points Two Clusters
• Can handle non-elliptical shapes
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Limitations of MIN
Original Points Two Clusters
• Sensitive to noise and outliers
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Cluster Similarity: MAX or Complete Linkage
Similarity of two clusters is based on the two least similar (most distant) points in the different clusters– Determined by all pairs of points in the two
clustersI1 I2 I3 I4 I5
I1 1.00 0.90 0.10 0.65 0.20I2 0.90 1.00 0.70 0.60 0.50I3 0.10 0.70 1.00 0.40 0.30I4 0.65 0.60 0.40 1.00 0.80I5 0.20 0.50 0.30 0.80 1.00 1 2 3 4 5
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Hierarchical Clustering: MAX
Nested Clusters Dendrogram
3 6 4 1 2 50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
1
2
3
4
5
61
2 5
3
4
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Strength of MAX
Original Points Two Clusters
• Less susceptible to noise and outliers
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Limitations of MAX
Original Points Two Clusters
• Tends to break large clusters• Biased towards globular clusters
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Cluster Similarity: Group Average
Proximity of two clusters is the average of pairwise proximity between points in the two clusters.
Need to use average connectivity for scalability since total proximity favors large clusters
||Cluster||Cluster
)p,pproximity(
)Cluster,Clusterproximity(ji
ClusterpClusterp
ji
jijjii
I1 I2 I3 I4 I5I1 1.00 0.90 0.10 0.65 0.20I2 0.90 1.00 0.70 0.60 0.50I3 0.10 0.70 1.00 0.40 0.30I4 0.65 0.60 0.40 1.00 0.80I5 0.20 0.50 0.30 0.80 1.00 1 2 3 4 5
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Hierarchical Clustering: Group Average
Nested Clusters Dendrogram
3 6 4 1 2 50
0.05
0.1
0.15
0.2
0.25
1
2
3
4
5
61
2
5
3
4
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Hierarchical Clustering: Group Average
Compromise between Single and Complete Link
Strengths– Less susceptible to noise and outliers
Limitations– Biased towards globular clusters
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Cluster Similarity: Ward’s Method
Similarity of two clusters is based on the increase in squared error when two clusters are merged– Similar to group average if distance between
points is distance squared
Less susceptible to noise and outliers
Biased towards globular clusters
Hierarchical analogue of K-means– Can be used to initialize K-means
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Hierarchical Clustering: Comparison
Group Average
Ward’s Method
1
23
4
5
61
2
5
3
4
MIN MAX
1
23
4
5
61
2
5
34
1
23
4
5
61
2 5
3
41
23
4
5
61
2
3
4
5
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Hierarchical Clustering: Time and Space requirements
O(N2) space since it uses the proximity matrix. – N is the number of points.
O(N3) time in many cases– There are N steps and at each step the size,
N2, proximity matrix must be updated and searched
– Complexity can be reduced to O(N2 log(N) ) time for some approaches
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Hierarchical Clustering: Problems and Limitations
Once a decision is made to combine two clusters, it cannot be undone
No objective function is directly minimized
Different schemes have problems with one or more of the following:– Sensitivity to noise and outliers– Difficulty handling different sized clusters and
convex shapes– Breaking large clusters
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Summary
Discussed hierarchical clustering, particularly the bottom-up approaches
Different ways to calculate intra-cluster distance/similarity for hierarchical clustering
Started to look at spectral clustering
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Spectral Clustering
Most popular modern clustering algorithms Simple to implement, can be solved by existing
software Not easy to see why it works Mathematically neat
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A Tutorial on Spectral Clustering\Arik Azran
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Spectral Clustering Example – 2 Spirals
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Dataset exhibits complex cluster shapesÞ K-means performs
very poorly in this space due to bias toward dense spherical clusters.
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0.709 -0.7085 -0.708 -0.7075 -0.707 -0.7065 -0.706In the embedded space given by two leading eigenvectors, clusters are trivial to separate.Spectral Clustering - Derek Greene
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Matthias Hein and Ulrike von Luxburg August 2007
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Algorithm
before
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Eigenvectors & Eigenvalues
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Algorithm
before
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Why?
If we eventually use K-means, why not just apply K-means to the original data?
This method allows us to cluster non-convex regions
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